1. Field of the Invention
This invention relates generally to data transmission systems, and more particularly, to a system of encoding, transmitting, and decoding data that is fast and reliable, and transmits data at rates near theoretical capacity along a noisy transmission medium.
2. Description of the Prior Art
Historically, in the analog communication era the FCC would allocate a predetermined bandwidth, i.e., frequency band, for transmission of information, illustratively music, over the air as the transmission medium. The signal typically took the form of a sinusoidal carrier wave that was modulated in response to the information. The modulation generally constituted analog modulation, where the amplitude of the carrier signal (AM) was varied in response to the information, or alternatively the frequency of the carrier signal was varied (FM) in response to the information desired to be transmitted. At the receiving end of the transmission, a receiver, typically a radio, consisting primarily of a demodulator, would produce a signal responsive to the amplitude or frequency of the modulated carrier, and eliminate the carrier itself. The received signal was intended to replicate the original information, yet was subjected to the noise of the transmission channel.
During the 1940s, a mathematical analysis was presented that shifted the thinking as to the manner by which information could reliably be transferred. Mathematician Claude Shannon introduced the communications model for coded information in which an encoder is introduced before any modulation and transmission and at the receiver a decoder is introduced after any demodulation. Shannon proved that rather than listening to noisy communication, the decoding end of the transmission could essentially recover the originally intended information, as if the noise were removed, even though the transmitted signal could not easily be distinguished in the presence of the noise. One example of this proof is in modern compact disc players where the music heard is essentially free of noise notwithstanding that the compact disc medium might have scratches or other defects.
In regard of the foregoing, Shannon identified two of the three significant elements associated with reliable communications. The first concerned the probability of error, whereby in an event of sufficient noise corruption, the information cannot be reproduced at the decoder. This established a need for a system wherein as code length increases, the probability of error decreases, preferably exponentially.
The second significant element of noise removal related to the rate of the communication, which is the ratio of the length of the original information to the length in its coded form. Shannon proved mathematically that there is a maximum rate of transmission for any given transmission medium, called the channel capacity C.
The standard model for noise is Gaussian. In the case of Gaussian noise, the maximum rate of the data channel between the encoder and the decoder as determined by Shannon corresponds to the relationship C=(½)log2(1+P/σ2), where P/σ2 corresponds to the signal-to-noise ratio (i.e., the ratio of the signal power to the noise power, where power is the average energy per transmitted symbol). Here the value P corresponds to a power constraint. More specifically, there is a constraint on the amount of energy that would be used during transmission of the information.
The third significant element associated with reliable communications is the code complexity, comprising encoder and decoder size (e.g., size of working memory and processors), the encoder and decoder computation time, and the time delay between sequences of source information and decoded information.
To date, no one other than the inventors herein has achieved a computationally feasible, mathematically proven scheme that achieves rates of transmission that are arbitrarily close to the Shannon capacity, while also achieving exponentially small error probability, for an additive noise channel.
Low density parity check (LDPC) codes and so called “turbo” codes were empirically demonstrated (in the 1990s) through simulations to achieve high rate and small error probability for a range of code sizes, that make these ubiquitous for current coded communications devices, but these codes lack demonstration of performance scalability for larger code sizes. It has not been proven mathematically that for rate near capacity, that these codes will achieve the low probabilities of error that will in the future be required of communications systems. An exception is the case of the comparatively simple erasure channel, which is not an additive noise channel.
Polarization codes, a more recent development of the last three years of a class of computationally feasible codes for channels with a finite input set, do achieve any rate below capacity, but the scaling of the error probability is not as effective.
There is therefore a need for a code system for communications that can be mathematically proven as well as empirically demonstrated to achieve the necessary low probabilities of error, high rate, and feasible complexity, for real-valued additive noise channels.
There is additionally a need for a code system that can mathematically be proven to be scalable wherein the probability of error decreases exponentially as the code word length is increased, with an acceptable scaling of complexity, and at rates of transmission that approach the Shannon capacity.